Bistable pulsating fronts in slowly oscillating environments *

Abstract

We consider reaction-diffusion fronts in spatially periodic bistable media with large periods. Whereas the homogenization regime associated with small periods had been well studied for bistable or Fisher-KPP reactions and, in the latter case, a formula for the limit minimal speeds of fronts in media with large periods had also been obtained thanks to the linear formulation of these minimal speeds and their monotonicity with respect to the period, the main remaining open question is concerned with fronts in bistable environments with large periods. In bistable media the unique front speeds are not linearly determined and are not monotone with respect to the spatial period in general, making the analysis of the limit of large periods more intricate. We show in this paper the existence of and an explicit formula for the limit of bistable front speeds as the spatial period goes to infinity. We also prove that the front profiles converge to a family of front profiles associated with spatially homogeneous equations. The main results are based on uniform estimates on the spatial width of the fronts, which themselves use zero number properties and intersection arguments.

Convergence to traveling waves for time-periodic bistable reaction-diffusion equations

“…The existence of pulsating waves for small or large T > 0, as well as the asymptotic limits of wave speed (denoted by e T ), were established in [7,8,9]. In particular, it is known that as T → 0 + , e T converges to the wave speed for the corresponding homogenized equation ([9, Theorems 1.2 and 1.4]), which is similar to our Theorem 1.3.…”

“…The proof uses as an essential ingredient the property that the wave profile Proposition 4.8 below). We point out that, a similar idea was used in a recent paper by the author and Hamel, Liang [7], which determined the asymptotic limits of wave speed and wave profile for bistable pulsating wave in slowly periodically oscillating media (see [7,Theorem 1.6]). A stronger assumption on f (f is homogeneous near u = 0 and u = 1) 5 was used in [7] to construct super-and sub-solution, while such an assumption is not needed in our theorem.…”

“…We point out that, a similar idea was used in a recent paper by the author and Hamel, Liang [7], which determined the asymptotic limits of wave speed and wave profile for bistable pulsating wave in slowly periodically oscillating media (see [7,Theorem 1.6]). A stronger assumption on f (f is homogeneous near u = 0 and u = 1) 5 was used in [7] to construct super-and sub-solution, while such an assumption is not needed in our theorem. Moreover, as we will discuss in Remark 1.8, our convergence result for temporally heterogeneous equation exhibits some specific features when compared to that for spatially periodic equation.…”

“…In Section 4.1, we collect some properties of (φ(•; g • s), c(g • s)) with respect to g ∈ H(f ) and s ∈ R, which will be used in the construction of super-and sub-solution for equation (1.1) when T is large in Section 4.2. It should be pointed that, in the case where f (s, u) is periodic in s and of bistable form in the sense of (1.23), the main results stated in Section 4.1 have been proved in [7,Proposition 4.1], but extra arguments are required to address the difficulties posed by the almost periodicity of f and the possible existence of multiple zeros of f (s, •).…”

“…where f (x, •) is periodic in x, the problem has been better understood. As a matter of fact, the limits of wave speeds for pulsating waves (which is an extension of traveling waves in spatially periodic media, analogously to time periodic traveling waves defined above) as T → 0 + or as T → +∞ were established for equations with various types of nonlinearties, and some interesting consequences regarding the effects of spatial heterogeneity were obtained, see e.g., [16,41,20,21,22,23,7,8,9,11]. Note that the aforementioned works are all concerned with periodic equations.…”